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Tangent to a Circle and Finding Line Intersections

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Tangent to a Circle and Finding Line Intersections
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Ceri Thomas

@cerithomas

·

10 Followers

Follow

A comprehensive guide to geometric concepts focusing on finding intersection of straight line graphs, equation of a tangent to a circle, and calculating perpendicular bisectors in geometry.

  • Explores fundamental concepts of straight line equations including gradients, y-intercepts, and parallel/perpendicular relationships
  • Details circle geometry including tangents, intersections, and center calculations
  • Covers advanced applications like finding circle equations from three points
  • Demonstrates practical problem-solving using simultaneous equations and geometric principles
  • Includes essential formulas for distance calculations and circle equations

29/04/2023

99

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Circle Properties and Equations

This section covers the fundamental properties of circles and their mathematical representations. The standard form equation (x-a)² + (y-b)² = r² is explored in detail.

Definition: A tangent is a line that touches the circle at exactly one point.

Vocabulary: The center coordinates are represented as (a,b), while r denotes the radius.

Example: When determining intersections between a line and circle, the discriminant (b²-4ac) indicates:

  • b²-4ac > 0: Two intersection points
  • b²-4ac = 0: One point (tangent)
  • b²-4ac < 0: No intersection
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Advanced Circle Applications

This section demonstrates practical applications of circle equations and intersection calculations. It includes detailed examples of finding intersection points and tangent equations.

Example: For the equation x² + (y-2)² = 29:

  1. Solve for intersection points using substitution
  2. Results in coordinates (2,7) and (-3,0)

Highlight: The perpendicular bisector of any chord always passes through the circle's center.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Circle Center Determination

This section explains methods for finding a circle's center using known points and perpendicular bisectors.

Definition: The perpendicular bisector method involves:

  1. Finding equations of perpendicular bisectors
  2. Calculating their intersection point
  3. This intersection determines the circle's center

Vocabulary: A diameter is any line segment passing through the center connecting two points on the circle.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Geometric Relationships and Applications

The final section synthesizes previous concepts to solve complex geometric problems involving circles and lines.

Highlight: The diameter (d) equals twice the radius (r), expressed as d = 2r.

Example: When finding a circle's equation from three points:

  1. Calculate perpendicular bisectors
  2. Find their intersection (center)
  3. Calculate radius using distance formula
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

View

Straight Line Graphs Fundamentals

This section introduces the core concepts of straight line equations and their relationships. The fundamental equation y = mx + c forms the basis for understanding line gradients and intersections.

Definition: The gradient (m) represents the slope of a line, calculated using the formula m = (y₂-y₁)/(x₂-x₁).

Vocabulary: Y-intercept (c) is the point where a line crosses the y-axis.

Example: The length of a line segment can be calculated using the formula: length = √((y₂-y₁)² + (x₂-x₁)²)

Highlight: For perpendicular lines, the product of their gradients (m₁ × m₂) equals -1.

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Know Higher Maths Straight Line thumbnail

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S5

Higher Maths Straight Line

Worked straight line problems with strategies

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Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

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Knowunity is the #1 education app in five European countries

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iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

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Lena, iOS user

I love this app ❤️ I actually use it every time I study.

Subjects

/

Maths

/

Pure maths

/

Coordinate geometry

Tangent to a Circle and Finding Line Intersections

user profile picture

Ceri Thomas

@cerithomas

·

10 Followers

Follow

A comprehensive guide to geometric concepts focusing on finding intersection of straight line graphs, equation of a tangent to a circle, and calculating perpendicular bisectors in geometry.

  • Explores fundamental concepts of straight line equations including gradients, y-intercepts, and parallel/perpendicular relationships
  • Details circle geometry including tangents, intersections, and center calculations
  • Covers advanced applications like finding circle equations from three points
  • Demonstrates practical problem-solving using simultaneous equations and geometric principles
  • Includes essential formulas for distance calculations and circle equations

29/04/2023

99

 

12

 

Maths

6

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Circle Properties and Equations

This section covers the fundamental properties of circles and their mathematical representations. The standard form equation (x-a)² + (y-b)² = r² is explored in detail.

Definition: A tangent is a line that touches the circle at exactly one point.

Vocabulary: The center coordinates are represented as (a,b), while r denotes the radius.

Example: When determining intersections between a line and circle, the discriminant (b²-4ac) indicates:

  • b²-4ac > 0: Two intersection points
  • b²-4ac = 0: One point (tangent)
  • b²-4ac < 0: No intersection
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Advanced Circle Applications

This section demonstrates practical applications of circle equations and intersection calculations. It includes detailed examples of finding intersection points and tangent equations.

Example: For the equation x² + (y-2)² = 29:

  1. Solve for intersection points using substitution
  2. Results in coordinates (2,7) and (-3,0)

Highlight: The perpendicular bisector of any chord always passes through the circle's center.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Circle Center Determination

This section explains methods for finding a circle's center using known points and perpendicular bisectors.

Definition: The perpendicular bisector method involves:

  1. Finding equations of perpendicular bisectors
  2. Calculating their intersection point
  3. This intersection determines the circle's center

Vocabulary: A diameter is any line segment passing through the center connecting two points on the circle.

straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Geometric Relationships and Applications

The final section synthesizes previous concepts to solve complex geometric problems involving circles and lines.

Highlight: The diameter (d) equals twice the radius (r), expressed as d = 2r.

Example: When finding a circle's equation from three points:

  1. Calculate perpendicular bisectors
  2. Find their intersection (center)
  3. Calculate radius using distance formula
straight line graphs y=MxC+c gradient M= y-intercept АУ y₂-y. Ax x₂-x, 8 length of a line = √ (y₂-y,)²+(₂-x,)² parallel lines У-у,=M(0-х,).

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Straight Line Graphs Fundamentals

This section introduces the core concepts of straight line equations and their relationships. The fundamental equation y = mx + c forms the basis for understanding line gradients and intersections.

Definition: The gradient (m) represents the slope of a line, calculated using the formula m = (y₂-y₁)/(x₂-x₁).

Vocabulary: Y-intercept (c) is the point where a line crosses the y-axis.

Example: The length of a line segment can be calculated using the formula: length = √((y₂-y₁)² + (x₂-x₁)²)

Highlight: For perpendicular lines, the product of their gradients (m₁ × m₂) equals -1.

Similar content

Maths - Higher Maths Straight Line

Worked straight line problems with strategies

4

104

0

Know Higher Maths Straight Line thumbnail

Can't find what you're looking for? Explore other subjects.

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the #1 education app in five European countries

4.9+

Average app rating

15 M

Pupils love Knowunity

#1

In education app charts in 12 countries

950 K+

Students have uploaded notes

Still not convinced? See what other students are saying...

iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

The app is very simple and well designed. So far I have always found everything I was looking for :D

Lena, iOS user

I love this app ❤️ I actually use it every time I study.